Intersections of SLE Paths: the double and cut point dimension of SLE

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• 作者：Jason Miller ; Hao Wu
• 关键词：Schramm–Loewner evolution (SLE) ; Hausdorff dimension ; Double points ; Cut points ; Gaussian free field (GFF) ; Imaginary geometry
• 刊名：Probability Theory and Related Fields
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：167
• 期：1-2
• 页码：45-105
• 全文大小：1808KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Probability Theory and Stochastic Processes; Theoretical, Mathematical and Computational Physics; Quantitative Finance; Mathematical and Computational Biology; Statistics for Business/Economics/Mathem
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-2064
• 卷排序：167

文摘

We compute the almost-sure Hausdorff dimension of the double points of chordal \(\mathrm {SLE}_\kappa \) for \(\kappa > 4\), confirming a prediction of Duplantier–Saleur (1989) for the contours of the FK model. We also compute the dimension of the cut points of chordal \(\mathrm {SLE}_\kappa \) for \(\kappa > 4\) as well as analogous dimensions for the radial and whole-plane \(\mathrm {SLE}_\kappa (\rho )\) processes for \(\kappa > 0\). We derive these facts as consequences of a more general result in which we compute the dimension of the intersection of two flow lines of the formal vector field \(e^{ih/\chi }\), where h is a Gaussian free field and \(\chi > 0\), of different angles with each other and with the domain boundary.